Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrng0 | Structured version Visualization version GIF version |
Description: The additive identity of R is the complex number 0. (Contributed by AV, 11-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) |
Ref | Expression |
---|---|
2zrng0 | ⊢ 0 = (0g‘𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncrng 20565 | . . 3 ⊢ ℂfld ∈ CRing | |
2 | crngring 19307 | . . 3 ⊢ (ℂfld ∈ CRing → ℂfld ∈ Ring) | |
3 | ringmnd 19305 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ ℂfld ∈ Mnd |
5 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
6 | 5 | 0even 44201 | . 2 ⊢ 0 ∈ 𝐸 |
7 | ssrab2 4055 | . . . 4 ⊢ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ⊆ ℤ | |
8 | 5, 7 | eqsstri 4000 | . . 3 ⊢ 𝐸 ⊆ ℤ |
9 | zsscn 11988 | . . 3 ⊢ ℤ ⊆ ℂ | |
10 | 8, 9 | sstri 3975 | . 2 ⊢ 𝐸 ⊆ ℂ |
11 | 2zrngbas.r | . . 3 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
12 | cnfldbas 20548 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
13 | cnfld0 20568 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
14 | 11, 12, 13 | ress0g 17938 | . 2 ⊢ ((ℂfld ∈ Mnd ∧ 0 ∈ 𝐸 ∧ 𝐸 ⊆ ℂ) → 0 = (0g‘𝑅)) |
15 | 4, 6, 10, 14 | mp3an 1457 | 1 ⊢ 0 = (0g‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {crab 3142 ⊆ wss 3935 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 · cmul 10541 2c2 11691 ℤcz 11980 ↾s cress 16483 0gc0g 16712 Mndcmnd 17910 Ringcrg 19296 CRingccrg 19297 ℂfldccnfld 20544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-cmn 18907 df-mgp 19239 df-ring 19298 df-cring 19299 df-cnfld 20545 |
This theorem is referenced by: 2zrngagrp 44213 |
Copyright terms: Public domain | W3C validator |